--- a/BAlgbra.agda	Sun Jul 05 15:49:00 2020 +0900
+++ b/BAlgbra.agda	Sun Jul 05 16:56:21 2020 +0900
@@ -19,60 +19,66 @@
 open OD O
 open OD.OD
 open ODAxiom odAxiom
+open HOD
 
 open _∧_
 open _∨_
 open Bool
 
 _∩_ : ( A B : HOD  ) → HOD
-A ∩ B = record { def = λ x → def A x ∧ def B x } 
+A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x } ;
+    odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y)) }
 
 _∪_ : ( A B : HOD  ) → HOD
-A ∪ B = record { def = λ x → def A x ∨ def B x } 
+A ∪ B = record { od = record { def = λ x → odef A x ∨ odef B x } ;
+    odmax = omax (odmax A) (odmax B) ; <odmax = lemma } where
+      lemma :  {y : Ordinal} → odef A y ∨ odef B y → y o< omax (odmax A) (odmax B)
+      lemma {y} (case1 a) = ordtrans (<odmax A a) (omax-x _ _)
+      lemma {y} (case2 b) = ordtrans (<odmax B b) (omax-y _ _)
 
 _＼_ : ( A B : HOD  ) → HOD
-A ＼ B = record { def = λ x → def A x ∧ ( ¬ ( def B x ) ) }
+A ＼ B = record { od = record { def = λ x → odef A x ∧ ( ¬ ( odef B x ) ) }; odmax = odmax A ; <odmax = λ y → <odmax A (proj1 y) }
 
 ∪-Union : { A B : HOD } → Union (A , B) ≡ ( A ∪ B )
 ∪-Union {A} {B} = ==→o≡ ( record { eq→ =  lemma1 ; eq← = lemma2 } )  where
-    lemma1 :  {x : Ordinal} → def (Union (A , B)) x → def (A ∪ B) x
+    lemma1 :  {x : Ordinal} → odef (Union (A , B)) x → odef (A ∪ B) x
     lemma1 {x} lt = lemma3 lt where
-        lemma4 : {y : Ordinal} → def (A , B) y ∧ def (ord→od y) x → ¬ (¬ ( def A x ∨ def B x) )
+        lemma4 : {y : Ordinal} → odef (A , B) y ∧ odef (ord→od y) x → ¬ (¬ ( odef A x ∨ odef B x) )
         lemma4 {y} z with proj1 z
-        lemma4 {y} z | case1 refl = double-neg (case1 ( subst (λ k → def k x ) oiso (proj2 z)) )
-        lemma4 {y} z | case2 refl = double-neg (case2 ( subst (λ k → def k x ) oiso (proj2 z)) )
-        lemma3 : (((u : Ordinals.ord O) → ¬ def (A , B) u ∧ def (ord→od u) x) → ⊥) → def (A ∪ B) x
+        lemma4 {y} z | case1 refl = double-neg (case1 ( subst (λ k → odef k x ) oiso (proj2 z)) )
+        lemma4 {y} z | case2 refl = double-neg (case2 ( subst (λ k → odef k x ) oiso (proj2 z)) )
+        lemma3 : (((u : Ordinals.ord O) → ¬ odef (A , B) u ∧ odef (ord→od u) x) → ⊥) → odef (A ∪ B) x
         lemma3 not = ODC.double-neg-eilm O (FExists _ lemma4 not)   -- choice
-    lemma2 :  {x : Ordinal} → def (A ∪ B) x → def (Union (A , B)) x
-    lemma2 {x} (case1 A∋x) = subst (λ k → def (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) A
-       (record { proj1 = case1 refl ; proj2 = subst (λ k → def A k) (sym diso) A∋x}))
-    lemma2 {x} (case2 B∋x) = subst (λ k → def (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) B
-       (record { proj1 = case2 refl ; proj2 = subst (λ k → def B k) (sym diso) B∋x}))
+    lemma2 :  {x : Ordinal} → odef (A ∪ B) x → odef (Union (A , B)) x
+    lemma2 {x} (case1 A∋x) = subst (λ k → odef (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) A
+       (record { proj1 = case1 refl ; proj2 = subst (λ k → odef A k) (sym diso) A∋x}))
+    lemma2 {x} (case2 B∋x) = subst (λ k → odef (Union (A , B)) k) diso ( IsZF.union→ isZF (A , B) (ord→od x) B
+       (record { proj1 = case2 refl ; proj2 = subst (λ k → odef B k) (sym diso) B∋x}))
 
 ∩-Select : { A B : HOD } →  Select A (  λ x → ( A ∋ x ) ∧ ( B ∋ x )  ) ≡ ( A ∩ B )
 ∩-Select {A} {B} = ==→o≡ ( record { eq→ =  lemma1 ; eq← = lemma2 } ) where
-    lemma1 : {x : Ordinal} → def (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x → def (A ∩ B) x
-    lemma1 {x} lt = record { proj1 = proj1 lt ; proj2 = subst (λ k → def B k ) diso (proj2 (proj2 lt)) }
-    lemma2 : {x : Ordinal} → def (A ∩ B) x → def (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x
+    lemma1 : {x : Ordinal} → odef (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x → odef (A ∩ B) x
+    lemma1 {x} lt = record { proj1 = proj1 lt ; proj2 = subst (λ k → odef B k ) diso (proj2 (proj2 lt)) }
+    lemma2 : {x : Ordinal} → odef (A ∩ B) x → odef (Select A (λ x₁ → (A ∋ x₁) ∧ (B ∋ x₁))) x
     lemma2 {x} lt = record { proj1 = proj1 lt ; proj2 =
-        record { proj1 = subst (λ k → def A k) (sym diso) (proj1 lt) ; proj2 = subst (λ k → def B k ) (sym diso) (proj2 lt) } }
+        record { proj1 = subst (λ k → odef A k) (sym diso) (proj1 lt) ; proj2 = subst (λ k → odef B k ) (sym diso) (proj2 lt) } }
 
 dist-ord : {p q r : HOD } → p ∩ ( q ∪ r ) ≡   ( p ∩ q ) ∪ ( p ∩ r )
 dist-ord {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
-    lemma1 :  {x : Ordinal} → def (p ∩ (q ∪ r)) x → def ((p ∩ q) ∪ (p ∩ r)) x
+    lemma1 :  {x : Ordinal} → odef (p ∩ (q ∪ r)) x → odef ((p ∩ q) ∪ (p ∩ r)) x
     lemma1 {x} lt with proj2 lt
     lemma1 {x} lt | case1 q∋x = case1 ( record { proj1 = proj1 lt ; proj2 = q∋x } )
     lemma1 {x} lt | case2 r∋x = case2 ( record { proj1 = proj1 lt ; proj2 = r∋x } )
-    lemma2  : {x : Ordinal} → def ((p ∩ q) ∪ (p ∩ r)) x → def (p ∩ (q ∪ r)) x
+    lemma2  : {x : Ordinal} → odef ((p ∩ q) ∪ (p ∩ r)) x → odef (p ∩ (q ∪ r)) x
     lemma2 {x} (case1 p∩q) = record { proj1 = proj1 p∩q ; proj2 = case1 (proj2 p∩q ) } 
     lemma2 {x} (case2 p∩r) = record { proj1 = proj1 p∩r ; proj2 = case2 (proj2 p∩r ) } 
 
 dist-ord2 : {p q r : HOD } → p ∪ ( q ∩ r ) ≡   ( p ∪ q ) ∩ ( p ∪ r )
 dist-ord2 {p} {q} {r} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
-    lemma1 : {x : Ordinal} → def (p ∪ (q ∩ r)) x → def ((p ∪ q) ∩ (p ∪ r)) x
+    lemma1 : {x : Ordinal} → odef (p ∪ (q ∩ r)) x → odef ((p ∪ q) ∩ (p ∪ r)) x
     lemma1 {x} (case1 cp) = record { proj1 = case1 cp ; proj2 = case1 cp }
     lemma1 {x} (case2 cqr) = record { proj1 = case2 (proj1 cqr) ; proj2 = case2 (proj2 cqr) }
-    lemma2 : {x : Ordinal} → def ((p ∪ q) ∩ (p ∪ r)) x → def (p ∪ (q ∩ r)) x
+    lemma2 : {x : Ordinal} → odef ((p ∪ q) ∩ (p ∪ r)) x → odef (p ∪ (q ∩ r)) x
     lemma2 {x} lt with proj1 lt | proj2 lt
     lemma2 {x} lt | case1 cp | _ = case1 cp
     lemma2 {x} lt | _ | case1 cp = case1 cp